In this section we show the first technical result of this paper: that the additive error \(\varepsilon \max _{p\in P}|f(p,q)|\)  in (REF ) can be replaced by \(\varepsilon \) , and the assumption \(\sum _{i=1}w_i=1\)  in Theorem REF  can be removed.
The condition for this to work is that the total importance value \(t\) , as defined below, is small. More precisely, the required sample size will be near-linear with \(t\) . Also, the uniform sampling in Theorem REF  will be replaced by non-uniform sampling with a distribution that is proportional to the importance of each point. This result is essentially a generalization and significant simplification of the framework in [1]} that used \(\varepsilon \) -approximations for constructing coresets. Maybe more importantly: using the idea of \((\varepsilon ,\nu )\) -approximations we are able to show that the sample size is near linear in \(t\)  while in [1]} it is quadratic in \(t\) . For some applications, such an improvement means turning a theoretical result into a practical result, especially when \(t\)  is close to \(\sqrt{n}\) .
